lighted rain behind a window
For the inner illumination of your drops remember to not make the drops 100% transparent. Look at the difference for value =1 and value=0.95


I don't think that there's any other option than doing this and adding a LOT of samples.. :(

You would have to modify the particles to be dynamic thought the flight, so their shape changes very slightly, so each moment they reflect light in a slightly other direction. 
Also, cycles sucks at caustics. Your lightning setup would use refraction for the drops themselves to be lit "from inside", and reflective to get the shimmering effect on the video.